I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots

For each $c$ and $p$ in
$\mathbb{R}^{n-1}$ such that $\|c-p\|
> < r$, there is an isotopy, $f_u^{c\;
> p} (\gamma), 0 \le u \le 1$, of
$\mathbb{R}^{n-1}$ onto itself which
transforms $c$ into $p$ and leaves
fixed all points of $\mathbb{R}^{n-1}$
outside the $(n-2)$-sphere of radius $r$